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Random fields on random graphs

Part of: Time-dependent statistical mechanics (dynamic and nonequilibrium) Special processes Graph theory

Published online by Cambridge University Press:  01 July 2016

P. Whittle
P. Whittle*
Affiliation:
University of Cambridge
*
∗ Postal address: Statistical Laboratory, University of Cambridge, 16 Mill Lane, Cambridge CB2 1SB, UK.
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Abstract

The distribution (1) used previously by the author to represent polymerisation of several types of unit also prescribes quite general statistics for a random field on a random graph. One has the integral expression (3) for its partition function, but the multiple complex form of the integral makes the nature of the expected saddlepoint evaluation in the thermodynamic limit unclear. It is shown in Section 4 that such an evaluation at a real positive saddlepoint holds, and subsidiary conditions narrowing down the choice of saddlepoint are deduced in Section 6. The analysis simplifies greatly in what is termed the semi-coupled case; see Sections 3, 5 and 7. In Section 8 the analysis is applied to an Ising model on a random graph of fixed degree r + 1. The Curie point of this model is found to agree with that deduced by Spitzer for an Ising model on an r-branching tree. This agreement strengthens the conclusion of ‘locally tree-like' behaviour of the graph, seen as an important property in a number of contexts.

MSC classification

Primary: 82C44: Dynamics of disordered systems (random Ising systems, etc.)
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 05C80: Random graphs
Type
Research Article
Information
Advances in Applied Probability , Volume 24 , Issue 2 , June 1992 , pp. 455 - 473
Copyright
Copyright © Applied Probability Trust 1992 

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